So I understand what rules you use where, and the general forms of the rules like:
My question is why are these the formulas that give us the answers we want? I learned integrals and derivatives using the limit method as the subdivisions got smaller and smaller, but I don't have a good conceptual understanding of the geometric manipulation taking place that allows the above rule to give us (slopes/areas) of exponential curves.
For example, I learned that when you integrate you're finding an bunch of infinitesimal area rectangles under the curve
where the height was the distance from the x-axis to the curve
so we get
and then to go from an infinitesimal area to the whole area you integrate to get
Now from here, if
And my question is why does changing the exponent of the curve in this way (add one, divide by new exponent) give us the geometric area under the curve?